Product of the off-diagonal block matrices

Question

Consider two real symmetric positive matrices $A\geq0$ and $B\geq 0$ with the following block form \begin{equation} A= \begin{pmatrix}\begin{array}{@{}c|c@{}} A_{11} & A_{12} \\ \hline A_{21} & A_{22} \\ \end{array}\end{pmatrix}, \quad B=\begin{pmatrix}\begin{array}{@{}c|c@{}} B_{11} & B_{12} \\ \hline B_{21} & B_{22} \\ \end{array}\end{pmatrix}. \end{equation} I need to know if $A_{12}B_{21}\geq 0$.
Thanks.